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Full Description
This book aims to present to first and second year graduate students a beautiful and relatively accessible field of mathematics-the theory of singu larities of stable differentiable mappings. The study of stable singularities is based on the now classical theories of Hassler Whitney, who determined the generic singularities (or lack of them) of Rn ‾ Rm (m ‾ 2n - 1) and R2 ‾ R2, and Marston Morse, for mappings who studied these singularities for Rn ‾ R. It was Rene Thorn who noticed (in the late '50's) that all of these results could be incorporated into one theory. The 1960 Bonn notes of Thom and Harold Levine (reprinted in [42]) gave the first general exposition of this theory. However, these notes preceded the work of Bernard Malgrange [23] on what is now known as the Malgrange Preparation Theorem-which allows the relatively easy computation of normal forms of stable singularities as well as the proof of the main theorem in the subject-and the definitive work of John Mather. More recently, two survey articles have appeared, by Arnold [4] and Wall [53], which have done much to codify the new material; still there is no totally accessible description of this subject for the beginning student. We hope that these notes will partially fill this gap. In writing this manuscript, we have repeatedly cribbed from the sources mentioned above-in particular, the Thom-Levine notes and the six basic papers by Mather.
Contents
I: Preliminaries on Manifolds.- §1. Manifolds.- §2. Differentiable Mappings and Submanifolds.- §3. Tangent Spaces.- §4. Partitions of Unity.- §5. Vector Bundles.- §6. Integration of Vector Fields.- II: Transversality.- §1. Sard's Theorem.- §2. Jet Bundles.- §3. The Whitney C? Topology.- §4. Transversality.- §5. The Whitney Embedding Theorem.- §6. Morse Theory.- §7. The Tubular Neighborhood Theorem.- III: Stable Mappings.- §1. Stable and Infinitesimally Stable Mappings.- §2. Examples.- §3. Immersions with Normal Crossings.- §4. Submersions with Folds.- IV: The Malgrange Preparation Theorem.- §1. The Weierstrass Preparation Theorem.- §2. The Malgrange Preparation Theorem.- §3. The Generalized Malgrange Preparation Theorem.- V: Various Equivalent Notions of Stability.- §1. Another Formulation of Infinitesimal Stability.- §2. Stability Under Deformations.- §3. A Characterization of Trivial Deformations.- §4. Infinitesimal Stability => Stability.- §5. Local Transverse Stability.- §6. Transverse Stability.- §7. Summary.- VI: Classification of Singularities, Part I: The Thom-Boardman Invariants.- §1. The Sr Classification.- §2. The Whitney Theory for Generic Mappings between 2-Manifolds.- §3. The Intrinsic Derivative.- §4. The Sr,s Singularities.- §5. The Thom-Boardman Stratification.- §6. Stable Maps Are Not Dense.- VII: Classification of Singularities, Part II: The Local Ring of a Singularity.- §1. Introduction.- §2. Finite Mappings.- §3. Contact Classes and Morin Singularities.- §4. Canonical Forms for Morin Singularities.- §5. Umbilics.- §6. Stable Mappings in Low Dimensions.- §A. Lie Groups.- Symbol Index.
